Article

Solution of Optimal Harvesting Problem by FiniteDifference Approximations of Size-StructuredPopulation Model

Johanna Pyy 1,* ID , Anssi Ahtikoski 2, Alexander Lapin 3 and Erkki Laitinen 1

1 Faculty of Sciences, University of Oulu, FI-90014 Oulu, Finland; erkki.laitinen@oulu.fi2 Natural Resources Institute Finland (Luke) Oulu, FI-90014 Oulu, Finland; anssi.ahtikoski@oulu.fi3 Institute of Computational Mathematics & Information Technology, Kazan Federal University,

420008 Kazan, Russia; avlapine@mail.ru* Correspondence: pyyj@student.oulu.fi

Received: 11 April 2018; Accepted: 24 April 2018 ; Published: 26 April 2018

Abstract: We solve numerically a forest management optimization problem governed by a nonlinearpartial differential equation (PDE), which is a size-structured population model. The formulatedproblem is supplemented with a natural constraint for a solution to be non-negative. PDE isapproximated by an explicit or implicit in time finite difference scheme, whereas the cost function istaken from the very beginning in the finite-dimensional form used in practice. We prove the stabilityof the constructed nonlinear finite difference schemes on the set of non-negative vectors and thesolvability of the formulated discrete optimal control problems. The gradient information is derivedby constructing discrete adjoint state equations. The projected gradient method is used for findingthe extremal points. The results of numerical testing for several real problems show good agreementwith the known results and confirm the theoretical statements.

Keywords: size-structured population model; nonlinear partial differential equation; finite differenceapproximation; optimization; gradient method

1. Introduction

The well-posedness of the continuous size-structured model has been studied in several papers(e.g., [1–4]). In Ref. [1], authors proved the local existence and uniqueness of a solution of the continuousmodel, where birth and mortality functions depend on total population. In Ref. [2], the authorsestablished the local existence and uniqueness of a solution of the size-structured nonlinear populationmodel, where also growth rate depends on total population. In the papers [3,4], the authors provedglobal existence and uniqueness of a solution of the continuous nonlinear population model, where allvital rates depend on total population. The total population can be described by e.g., total number ofindividuals (e.g., [3]), total biomass (e.g., [5]) or basal area.

A continuous nonlinear size-structured population model has been used in a forest managementoptimization problem (e.g., [6–8]). In a continuous formulation, this nonlinear optimization problemcannot be solved by analytic methods. A natural approach is to solve this problem by approximatinga continuous model by a discrete one and further solving a discrete optimization problem by iterativealgorithms. In this paper, we focus on development of finite difference schemes to approximate thesolution of a continuous nonlinear population model. Efficient schemes are essential for solvingoptimal control problems or parameter estimation problems as such problems require solving themodel numerous times before an optimal solution is obtained.

When continuous population model is approximated by a finite difference scheme, it becomesa matrix population model [9]. In matrix models, trees are divided into classes with respect to their

Math. Comput. Appl. 2018, 23, 22; doi:10.3390/mca23020022 www.mdpi.com/journal/mca

Math. Comput. Appl. 2018, 23, 22 2 of 15

size—for instance, diameter. The matrix describes how the class division changes at one time step.Matrix population models have also been used for forest management optimizations (e.g., [10,11]).

In optimization, using iterative algorithms is inevitable. Higher-order algorithms are usuallysensitive to the regularity of the solution, and, therefore, they usually yield a convergence rate offirst order as soon as the compatibility conditions are not satisfied. Moreover, in practice, the vitalrates are determined on a statistical basis and the compatibility conditions required for high-orderconvergence are hardly valid with real-life data. These suggest that, in most cases, a first-order methodshould be the most adequate. Hence, it is desirable to have a robust scheme that can produce manyuseful qualitative and quantitative properties of the solutions of the differential problem but requiresminimum regularity of the solution [12]. Unfortunately, one could not derive the explicit formula forthe optimal strategy since the strategy, the state and the costate are coupled into a complex system.The results at this stage may be regarded as a middle step to real world applications and serve asa starting point for numerical computations [13].

In our knowledge, the comprehensive theoretical investigation of the forest managementoptimization problem with a continuous nonlinear population model as the state equation is stilllacking, and, in that sense, the problem is an open problem. Hence, in this work, the investigation ofthe problem in its differential form is omitted, whereas, we consider the finite dimensional counterpartof the problem constructed by finite difference approximations of the state problem and taking forcost function a finite dimensional form used in practice. We prove the stability of constructed finitedifference schemes on the set of non-negative solutions and solvability of the optimization problem,and deduce the necessary gradient information for iterative solution methods. We solve severalapplied problems, where different approximations schemes are used, and compare the computedresults. The rest of the article is organized as follows. In Section 2, a mathematical model of optimalharvesting problem for the size-structured forest is formulated. In Section 3, we construct andinvestigate two finite difference approximation schemes for a nonlinear boundary value problemthat simulates the growth and the harvesting of a forest. A gradient method for minimizing the costfunction is constructed in Section 4. The theoretical details of this method are set out in the Appendix Ato the article. Section 5 is devoted to the numerical solution of a real-life problem and comparativeanalysis of the computing results. Finally, in Section 6, we present discussions.

2. Formulation of the Optimal Control Problem

In order to formulate the mathematical model for the optimal harvesting problem for thesize-structured forest, we define the following notations. In space, we denote by x ∈ Ω := (L0, L] thethickness of the tree, where L0 and L are the lower and upper bounds of the space domain, respectively.Moreover, t ∈ (0, T] is the time, where T is the upper limit. By Q, we denote the product spaceΩ× (0, T]. We denote by y(x, t) and h(x, t) the number of trees per unit area (state) and the number ofremoved trees per unit area (control), respectively. Now, the optimal harvesting problem where thecost functional J(y, h) characterizes a net present value (NPV) of ongoing rotation, and d(x, t) is thediscounted price function, is formulated as follows:

max(y,h)∈K

J(y, h) :=∫

Qd(x, t)h(x, t). (1)

Above K = Yad ×Had is the set of constraints for the state and the control, where

Yad =y | for all (x, t) ∈ Q : y(x, t) ≥ 0; y is a solution for Equations (4)–(6), (2)

.Had =h | for all (x, t) ∈ Q : 0 6 h(x, t) 6 hmax, for all t ∈ (0, T] :∫

Ωh(x, t) > B or

∫Ω

h(x, t) = 0 (3)

From the point of real-life problems, it is obvious that there exist constants hmax > 0 and B,which denote the upper limit for harvesting and lower limit for making profitable thinning of trees at

Math. Comput. Appl. 2018, 23, 22 3 of 15

time event t; otherwise, the thinning is not done. Notice that harvesting h depends on the state y (viaconstraint sets), which is defined by the population model

∂y(x, t)∂t

+∂(g(x, P(t))y(x, t))

∂x+ m(x, P(t))y(x, t) + h(x, t) = 0, in Q, (4)

g(0, P(t))y(0, t) = 0, in (0, T], (5)

y(x, 0) = y0(x), in Ω, (6)

where g(x, P(t)) is growth rate, m(x, P(t)) mortality rate and y0(x) > 0 is initial diameter distributionof the trees. Growth and mortality rates depend on diameter x of a tree and on the basal area, P(t),of the forest stand, where

P(t) = π∫ L

0

( x2

)2y(x, t)dx.

In the case h = 0, the problems (4)–(6) are a particular case of the problem that have beeninvestigated in [1–4]. In these articles, the existence of a non-negative continuous solution of thisproblem has been proved under some "natural" assumptions for input data. They are:

1. g(x, P) is continuous and strictly positive for all x and P and continuously differentiable withrespect to x;

2. m(x, P) is non-negative for x and P and integrable in x;3. g(x, P) and m(x, P) are Lipschitzian with respect to P;4. sup

x,Pm(x, P) < ∞.

We also assume that these assumptions are satisfied. We use growth rate g and mortality m ina bilinear form

g(x, P(t)) = g11 + g12x + (g21 + g22x)P(t),

m(x, P(t)) = m11 +m12

x+

m13

x2 +(

m21 +m22

x+

m23

x2

)P(t),

where the constants gij and mij are such that g(x, P) > 0 and m(x, P) > 0 for all x ∈ Ω and P > 0.Obviously, because of suppositions g(0, P(t)) > 0, the boundary condition (5) reads as y(0, t) = 0.

The optimal harvesting problem has been investigated in [6–8]. The authors of these publicationsconsidered the case where the harvesting function has the form h(x, t) = c(x, t)y(x, t), where c(x, t)is the control. Thus, they investigated a coefficient identification problem while we solve an optimalcontrol problem with distributed (on the right-hand side) control.

3. Finite Difference Approximations

In this chapter, we derive explicit and (semi)implicit finite difference approximations for the stateproblems (4)–(6) and prove their stability estimates on non-negative solutions. The investigation ofexistence, uniqueness and convergence of approximations is beyond the scope of our article. For thesize-structured population model with recruitment, the existence, uniqueness and convergence ofexplicit approximations is investigated in [14] and implicit approximation in [5,15].

The following notations are used throughout the paper: ∆t = TM and ∆x = L−L0

N denote thetemporal and spatial mesh size, respectively. The non-overlapping mesh intervals are (tk−1, tk],k = 1, . . . , M, and (xi−1, xi], i = 1, . . . , N, where t0 = 0, tM = T, x0 = L0, xN = L.

Let us denote by yki and hk

i the finite difference approximations of y(xi, tk) and h(xi, tk),respectively. Moreover, we denote gk

i := g(xi, Pk) and mki := m(xi, Pk) the discrete values of the

growth rate and mortality rate, respectively, in size class [xi−1, xi]. The discretized value of the basal

area at time tk is Pk := πN

∑i=1

(xi2)2y(xi, tk).

Math. Comput. Appl. 2018, 23, 22 4 of 15

3.1. Explicit Approximation of the State Equation

For all meshpoints i = 1, . . . , N; k = 1, . . . , M, the explicit finite difference approximation of thesize-structured population model (4)–(6) reads

yki − yk−1

i∆t

+gk−1

i yk−1i − gk−1

i−1 yk−1i−1

∆x+ mk−1

i yk−1i + hk

i = 0,

yk0 = 0, (7)

y0i > 0 constant.

Note that we use so-called upwind approximation for the first order derivative in space (variablex) using the positivity of coefficient g(x, P) on the set of non-negative mesh functions y. The explicitscheme (7) can be written in the form:

yki −

(1− ∆t

∆xgk−1

i − ∆tmk−1i

)yk−1

i −(

∆t∆x

gk−1i−1

)yk−1

i−1 + ∆thki = 0.

Later on, we denote by aki = 1 − ∆t

∆xgk

i − ∆tmki , and, bk

i =∆t∆x

gki . Moreover, we denote by

yk := (yk1, . . . , yk

N), hk := (hki , . . . , hk

N) the vectors of the nodal values and by

Ak =

ak

1 0 0 . . . 0 0bk

1 ak2 0 . . . 0 0

0 bk2 ak

3 . . . 0 0...

.... . . . . .

......

0 0 0 . . . bkN−1 ak

N

the matrix of coefficients. Now, we can write explicit difference scheme (7) in the followingalgebraic form:

yk −Ak−1yk−1 + ∆thk = 0, k = 1, . . . , M. (8)

Note that this scheme is just the forest growth model studied in [11]. Moreover, the numericalcalculation of the next temporal state involves only matrix to vector calculations. The drawback of theexplicit scheme is that the following stability condition (9) must be satisfied.

Lemma 1. Let the condition∆x > ∆t sup

x,tg(x, P(t)) (9)

be satisfied. Then, on the set of non-negative mesh functions y, the finite difference scheme (7) is stable

maxk‖yk‖1 6 C(T)

(‖y0‖1 +

M

∑k=1

∆t‖hk‖1), (10)

where ‖v‖1 =N∑

i=1|vi|.

Proof. On the non-negative mesh functions y, the coefficients gi(P) are positive and mi(P) > 0. For themesh steps satisfying condition (9), the diagonal entries of matrix Ak satisfy the inequality

|aki | 6 1− ∆t

∆xgi(Pk) + ∆tmi(Pk).

Math. Comput. Appl. 2018, 23, 22 5 of 15

Because of this inequality, we have the following estimate for ‖.‖1-norm of matrices, connectedwith ‖.‖1-norm of vectors:

‖Ak‖1 =N

∑i=1

(|aki |+ |bk

i |) 6 1 + Cm∆t

with Cm = supx,t m(x, P(t)). Due to this estimate and condition (9), we obtain from Equation (8)the inequality

‖yk‖1 6 (1 + Cm∆t)‖yk−1‖1 + ∆t‖hk‖1 for all k = 1, 2, . . . , M,

whence stability estimate (10) follows.

The condition (9) means that the length of the time step ∆t and width of the size class ∆x have tobe chosen so that a tree cannot grow over one size class during one time step ∆t (compare with [16]).

3.2. Implicit Approximation of the State Equation

For all meshpoints i = 1, . . . , N; k = 1, . . . , M, the implicit finite difference approximation ofthe models (4)–(6) is the following linearized problem, with nonlinear coefficients calculated on theprevious time level:

yki − yk−1

i∆t

+gk−1

i yki − gk−1

i−1 yki−1

∆x+ mk−1

i yki + hk

i = 0,

yk0 = 0, (11)

y0i > 0 constant.

Equation (11) can be rewritten as(1 +

∆t∆x

gk−1i + ∆tmk−1

i

)yk

i −∆t∆x

gk−1i−1 yk

i−1 − yk−1i + ∆thk

i = 0,

for all i = 1, . . . , N, k = 1, . . . , M.

Using the notations aki = 1 +

∆t∆x

gki + ∆tmk

i and bki = − ∆t

∆xgk

i , we rewrite Equation (11) in a formof linear algebraic equations

Bk−1yk − yk−1 + ∆thk = 0, k = 1, . . . , M, (12)

where

Bk :=

ak

1 0 0 . . . 0 0bk

1 ak2 0 . . . 0 0

0 bk2 ak

3 . . . 0 0...

.... . . . . .

......

0 0 0 . . . bkN−1 ak

N

is a matrix of nonlinear coefficients.

Lemma 2. Finite difference scheme (11) is unconditionally stable on the set of non-negative mesh functions y:for any ∆t and ∆x the following stability estimate holds:

maxk‖yk‖1 6 ‖y0‖1 +

M

∑k=1

∆t‖hk‖1. (13)

Math. Comput. Appl. 2018, 23, 22 6 of 15

Proof. By direct calculations, we obtain from Equation (12) the equality

N

∑i=1

aki yk

i +N−1

∑i=1

bki yk

i −N

∑i=1

yk−1i + ∆t

N

∑i=1

hki = 0.

Since aki + bk

i > 1 and akN > 1, then, from this equality, we get

N

∑i=1

yki 6

N

∑i=1

yk−1i + ∆t

N

∑i=1|hk

i |.

Because of positivity of vectors yk and yk−1, the last inequality can be written in the form

‖yk‖1 6 ‖yk−1‖1 + ∆t‖hk‖1 ∀k,

whence stability estimate (13) follows.

Notice, contrary to the explicit scheme the time step ∆t and class width ∆x has no mutualdependence, hence the growth of a tree during a time step is not restricted less than one size class.This characteristic of the implicit scheme is useful in the optimal harvesting problem, covered by themodels (4)–(6) or parameter identification problem because such problems require solving the modelmany times before an optimal solution is obtained.

3.3. Approximation of the Optimal Control Problem

We denote dki := d(xi, tk) the discounted price for size class (xi−1, xi] at time tk, and dk = (dk

1, . . . , dkN).

Moreover, (u, v) := ∑N1 uivi is the vector product of vectors u, v ∈ RN. Approximating the cost function (1)

by the right-hand Riemann sum, we get the following approximation for the harvesting problem:

max(y,h)∈K

J(y, h) :=

M

∑k=1

(dk, hk) =M

∑k=1

N

∑i=1

dki hk

i

. (14)

Above, we denote by K = Yad × Had, where

Yad =(y, h) |y > 0, y is a solution for Equations (7) or (11), (15)

Had =h | 0 6 hk 6 hmax, ‖hk‖1 > B or hk = 0, k = 1, . . . , M. (16)

Moreover, y = (y1, . . . , yM) and h = (h1, . . . , hM).The following propositions show that the discrete optimal harvesting problem (14) has at least

one solution in both cases, i.e., if models (4)–(6) is approximated explicitly or implicitly.

Proposition 1. Let the mesh steps ∆t and ∆x satisfy the inequality

1− ∆t∆x

supx,t

g(x, P(t))− ∆t supx,t

m(x, P(t)) > 0. (17)

Then, Problem (14) has at least one solution if y satisfies Equation (7).

Proof. The set K is non-empty. In fact, due to assumption (17), the solution y of finite differencescheme (7) with y0 > 0 is non-negative if h = 0. This statement can be easily verified using form (8) ofthe difference scheme and noting that all entries ak

i and bki of the matrices Ak are non-negative.

Obviously, assumption (9) follows from inequality (17), so stability estimate (10) holds. Sincevector h ∈ K is bounded, then, due to inequality (10), there exists a constant Y such that ‖y‖1 6 Y, i.e.,the set K is bounded. It is closed because of the continuity of functions g(P) and m(P) with respect

Math. Comput. Appl. 2018, 23, 22 7 of 15

to P, while P is obviously continuous with respect to y. Thus, K is compact. At last, cost functionJ of Problem (14) is continuous, whence the existence of a solution to Problem (14) follows fromWeierstrass’s theorem.

Proposition 2. Problem (14) has at least one solution if y satisfies Equation (11).

Proof. Proof is very similar to the proof of Proposition 1. Namely, the set K is non-empty because,for h = 0, the solution y of finite difference scheme (11) is non-negative for all ∆x and ∆t. Since h isbounded, then, due to stability estimate (13), y is also bounded, so the set K is bounded. It is closedbecause of the continuity of functions g(P) and m(P) with respect to P, and continuity of P withrespect to y. Thus, K is compact. At last, cost function J of Problem (14) is continuous, whence theexistence of a solution to Problem (14) follows.

Remark 1. Since neither the function J is strictly concave nor the set K is strictly convex, the optimizationproblem can have a non-unique solution.

4. Realization of the Optimal Strategies

In this section, a first order method to approximate the optimal harvesting problem (14) isconstructed. In real-life applications, the growth rate g and mortality rate m are determined ona statistical basis and the compatibility conditions required for high-order methods can be hardlyvalidated. Hence, a first order method, which is desirable to have a robust scheme but requiresminimum regularity of the solution should be the most adequate. The first order methods requirecomputing of the Fréchet derivatives (Jacobian matrix), which can be computationally expensive.However, when we consider the nonlinear optimization problem, only the gradient of the objectfunction is needed, and the gradient can be computed without the Fréchet derivatives. In this work,the adjoint approach developed in the 1970s in [17] is applied for calculation of the functional gradient.The adjoint method has a great advantage against the direct method because only one linear stateproblem, so called adjoint state, need to be solved for obtaining the gradient information. Today, it isa well-known method for computing the gradient of a functional with respect to model parameterswhen this functional depends on those model parameters trough state variables, which are solutionsof the state problem. However, this method is less well understood in the control of populationmodels, and, as far as we know, no applications to distributed optimal control of harvesting ispresented in literature. Duality and adjoint equations are essential tools in studying existence ofthe optimal pair (y, h), and, for a periodic age-dependent harvesting problem and for age-spatialstructured harvesting problem, it is applied for proving the existence of the bang-bang control in [18]and in [19,20], respectively. For continuous size-structured harvesting, problem duality and adjointequations are applied for proving the existence of the bang-bang control in [6,8].

In this work, we apply the Lagrange method and give a recipe to systematically define theadjoint state equations and gradient information. We formulate the Lagrangian of the problem (14)with respect to the state constraint (15) only, and use the projection method regarding the controlconstraint (16). In the projection method, if solution goes outside the constraint set (16), it is projectedback to there. Let us generalize and denote by A(y, h) = 0 the operator Equation (8) (or Equation (12)).Moreover, Ak(y, h) = 0 is the operator equation at the time level k, k = 1, . . . , M.

Suppose the functional J and operator A to be differentiable in the sense that there exist thefollowing partial derivatives:

Jyδy = limt→0

J(y + tδy, h)− J(y, h)t

, Jhδh = limt→0

J(y, h + tδh)− J(y, h)t

,

Ayδy = limt→0

A(y + tδy, h)− A(y, h)t

, Ahδh = limt→0

A(y, h + tδh)− A(y, h)t

Math. Comput. Appl. 2018, 23, 22 8 of 15

for all vectors δy and δh (or at least for the vectors such that y, y + δy ∈ Yad and h, h + δh ∈ Had).Note that for the fixed h and y, Jy ≡ Jy(y, h) and Jh ≡ Jh(y, h) are vectors while Ay = Ay(y, h) andAh = Ah(y, h) are matrices. By A∗y and A∗h, we denote the corresponding transpose matrices.

Let us define Lagrange function, L, of the problem (14) by

L(y, h, λ) = J(y(h), h)−M

∑k=1

(λk, Ak(y, h)),

where λk ∈ RN . Now, for all feasible pair (y, h) holds A(y, h) = 0, and, for any λ, we have:

L(y, h, λ) = J(y(h), h),

and, since λ does not depend on h, we have

∂ J∂h

=∂L(y, h, λ)

∂y∂y∂h

+∂L(y, h, λ)

∂h. (18)

Above, one method to approximate∂y∂h

is to compute N finite differences over control variable h.

However, each computation requires solving the equation A(y, h) = 0, and, for large N, this method

is computationally expensive. In the adjoint method, we can avoid to compute∂y∂h

by solving the linearadjoint state equation only once.

The theory of constrained optimization, see [21], says that (y, h) is the optimal pair for theproblem (14) if (y, h, λ) is a saddle point of L. The derivatives of L with respect to y, h and λ are:

∂L(y, h, λ)

∂y=

∂ J(y, h)∂y

−(∂A(y, h)

∂y

)∗λ,

∂L(y, h, λ)

∂h=

∂ J(y, h)∂h

−(∂A(y, h)

∂h

)∗λ,

∂L(y, h, λ)

∂λ= −A(y, h).

Now,∂L(y, h, λ)

∂λ= 0 gives the state equation,

∂L(y, h, λ)

∂y= 0 gives the adjoint state equation

and∂ J∂h

=∂L(y, h, λ)

∂hgives the gradient.

Now, the calculation of the gradient can be summarized by the following steps when theLagrangian L(y, h, λ) = J(y, h)− (λ, A(y, h)) is first formulated:

I Solve the state equation A(y, h) = 0;II Solve the adjoint state equation

∂L(y, h, λ)

∂y=

∂ J(y, h)∂y

−(∂A(y, h)

∂y

)∗λ = 0;

III Compute the gradient

∂ J∂h

=∂ J(y, h)

∂h−(∂A(y, h)

∂h

)∗λ.

Partial derivatives of J(y, h) and A(y, h) are presented in Appendix A. Gradient∂ J∂h

we usedin projected gradient method [22], which we applied for iteration of a solution of the optimalharvesting problem.

Math. Comput. Appl. 2018, 23, 22 9 of 15

5. Numerical Example

In this section, we study numerical examples of problem (14). We compared two cases where thestate constraint (4) was approximated with explicit approximation (7) and implicit approximation (11).

As the discounted price for size class (xi−1, xi] at time tk, we used dki =

cpvpi + csvs

i

(1 + r)tk , where r is

the interest rate, cp and cs are the prices of the pulpwood and sawlog, respectively, and vpi and

vsi are the volumes of pulpwood and sawlog of a tree in size class (xi−1, xi], respectively. In the

optimizations, we used the following values for parameters: price of pulpwood cp = 16.56 em−3 andsawlog cs = 58.44 em−3, interest rate r = 3% and lower bound for harvested trees B = 50 m3 ha−1.The pulpwood and sawlog volumes vp

i and vsi we got from [10]. The optimization results of problem (14)

are presented in Tables 1 and 2, and in Figures 1 and 2.

Figure 1. Diameter distributions associated with optimal thinnings of problem (14) with state equationapproximated by explicit scheme (7). Numbers (e.g., 5–8, 11–14) represent diameter in centimetres.

Math. Comput. Appl. 2018, 23, 22 10 of 15

Figure 2. Diameter distributions associated with optimal thinnings of problem (14) with state equationapproximated by implicit scheme (11). Numbers (e.g., 5–8, 11–14) represent diameter in centimetres.

Table 1. Maximum net present values, MaxNPVs, (i.e., optimal cost function values of the problem (14))and mean annual increments (MAI) associated with optimal stand-level managements. Initial densityis 1000 stems ha−1.

MaxNPV (eha−1) MAI (m3 ha−1 a−1)Explicit Implicit Explicit Implicit

Time step ∆t = 5 years and classwidth ∆x = 3 cm

4095 3996 2.94 2.83

Time step ∆t = 3 years and classwidth ∆x = 3 cm

4301 4231 2.94 3.11

Time step ∆t = 3 years and classwidth ∆x = 2 cm

4676 4648 2.96 3.00

Time step ∆t = 2 years and classwidth ∆x = 3 cm

4384 4311 2.93 3.04

Time step ∆t = 2 years and classwidth ∆x = 2 cm

4776 4726 3.09 3.09

Math. Comput. Appl. 2018, 23, 22 11 of 15

Table 2. Optimal stand-level managements generated by explicit (7) and implicit (11) approximationof the forest growth model (4). Initial density is 1000 stems ha−1.

Stand Age (a) Removal(m3 ha−1)

Thinning Intensity (% ofBasal Area Removed)

Saw LogProportion (%)

Explicit Implicit Explicit Implicit Explicit Implicit Explicit Implicit

Time step ∆t = 5 years and class width ∆x = 3 cm1st thinning 55 45 91.0 77.2 48 48 87 852nd thinning 70 60 68.9 59.3 46 46 88 873rd thinning 85 80 98.6 90.6 79 72 87 86final thinning 105 105 50.0 69.9 100 100 90 86total 308.5 297.0 88 86

Time step ∆t = 3 years and class width ∆x = 3 cm1st thinning 49 43 78.9 52.4 48 35 83 862nd thinning 64 58 68.2 54.3 50 35 86 893rd thinning 76 70 55.4 50.0 56 35 87 914th thinning - 82 - 72.4 - 59 - 86final thinning 100 100 91.5 82.2 100 100 88 85total 293.9 311.3 86 87

Time step ∆t = 3 years and class width ∆x = 2 cm1st thinning 46 40 60.6 60.5 36 40 83 812nd thinning 58 52 56.5 51.8 38 40 85 833rd thinning 70 64 59.7 52.1 45 46 88 864th thinning 79 79 50.0 55.0 52 56 89 90final thinning 94 97 72.6 71.4 100 100 89 86total 299.5 290.7 87 85

Time step ∆t = 2 years and class width ∆x = 3 cm1st thinning 47 45 62.2 59.1 39 38 84 862nd thinning 61 59 59.9 54.8 41 37 87 883rd thinning 75 71 58.9 50.0 44 37 90 904th thinning 83 85 64.4 78.0 70 64 84 87final thinning 101 103 50.0 71.0 100 100 86 86total 295.5 313.3 86 87

Time step ∆t = 2 years and class width ∆x = 2 cm1st thinning 43 41 55.3 58.5 36 38 80 822nd thinning 55 53 55.6 53.3 39 39 84 843rd thinning 65 65 50.0 52.0 42 44 86 874th thinning 77 77 53.0 50.2 53 51 89 89final thinning 95 95 79.5 79.2 100 100 89 87total 293.4 293.1 86 86

The results show that the maximum net present value (NPV) associated with the explicitapproximation (7) was higher than the corresponding of the implicit approximation (Table 1).When class width or time step decreased, maximum NPV increased in both cases. The difference ofmaximum NPVs between the two cases decreased when class width or time step decreased. Only whentime step decreased from three to two years, the difference of maximum NPVs increased. The differencewas biggest (99 e) when time step ∆t = 5 years and class width ∆x = 3 cm and smallest (28 e) whentime step ∆t = 3 years and class width ∆x = 2 cm.

With both approximations, three or four intermediate thinnings were made (Table 2). Numberof thinnings increased when time step and class width decreased. When implicit approximation (11)was used, first thinnings were made 1–2 time steps earlier, while the last few thinnings were made0–2 time steps later than when explicit approximation (7) was used. The thinning intensities werealmost identical between the two approximations. If there was some difference, intensity was usuallybigger when explicit approximation (7) was used (Table 2). The thinning pattern was in all optimalmanagements quite similar: in each thinning, more big trees than small ones were removed indicatinga thinning from-above method (for different thinning types, see e.g., [23], pp. 727, 733). Thinning from

Math. Comput. Appl. 2018, 23, 22 12 of 15

above has proven to be the best thinning type in stand-level optimizations of even-aged boreal forests(e.g., [24]). When explicit approximation (7) was used, all trees from two or three of the biggest sizeclasses were removed (Figure 1). On the other hand, when implicit approximation (11) was applied,only part of the trees from those size classes were removed (Figure 2).

6. Discussion

This study contributes to existing literature on forest management by providing a theoreticallysound framework to solve nonlinear optimization problem of even-aged stands. We compared theresults of forest management optimizations, when the explicit and implicit approximations of the forestgrowth model was used. The optimization results show that the differences of the results betweenapproximations are diminutive. This was expected as solutions of both approximation equations areproved to converge to solutions of continuous equation [5,14].

In numerical examples, we used data from the Scots pine (Pinus sylvestris L.) stands that werelocated in Northern Ostrobothnia, Finland, on nutrient-poor soil type. The data was the same asin [11]. The difference is that, in [11], data was fitted directly to the matrix model, as, in this study,we first fitted data to the continuous model and then approximated it with a matrix model. In [11],the time step was five years and class width 3 cm. The results are in line with each other. Both methodsgave four thinnings in optimal management and thinning from above dominated as the thinningtype. In [11], the optimal net present value was slightly higher and, in the optimal management,the thinnings were made slightly earlier than in this study.

The optimal harvesting problem with a continuous size-structured population model was studiedin [6–8]. In those papers, harvesting was defined as a proportion of removed trees. The maximumprinciple for the problem was proved in [6,8]. Moreover, in [7], the strong bang-bang principle undersome additional (but realistic) conditions was proved. This means that the optimal solution has thestructure, where all trees bigger than some certain size are removed. In our results, the solution ofthe optimization problem, where state constraint was approximated with explicit approximation,was nearer that structure. In addition, the optimization results were a little better then. However,when explicit approximation is used, the time step and class width have to be chosen so thata tree cannot grow over one size class during one time step [16]. We proved that only then is theexplicit approximation scheme stable. For the implicit approximation scheme, we proved that it isunconditionally stable. Thus, in implicit approximation, the time step and class width can be chosenfreely. In general, explicit approximation of the population model is more commonly used as a forestgrowth model [9,16].

Author Contributions: J.P. made the approximations and conducted the optimizations under supervision of E.L.;A.L. proved the theoretical results; J.P. and A.A. analyzed the numerical results; all authors contributed to thewriting of the manuscript.

Acknowledgments: We want to acknowledge the Jenny and Antti Wihuri foundation for financial support.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

We used the adjoint method to solve the optimization problem (14). For that, we needed thepartial derivatives of the Lagrangian

L(y, h, λ) = J(y, h)−M

∑k=1

(λk, Ak(y, h)),

where J(h) = ∑Mk=1(d

k, hk) is cost function of problem (14) and Ak(y, h) is constraint (8) or (12).First, we calculate the partial derivatives of the cost function J(y, h). Since it depends only on h,

obviously∂ J∂y

= 0. The partial derivative of the cost function J with respect to h is

Math. Comput. Appl. 2018, 23, 22 13 of 15

∂ J(h)∂h

= d.

Next, we calculate the partial derivatives of constraint function A(y, h). In both forms of A

(constraints (8) or (12)), the partial derivative with respect to h is∂A(y, h)

∂h= ∆t.

Let us calculate the partial derivative of constraint function (8) (explicit approximation of the stateEquation (4)) with respect to y

∂(λ, A(y, h))∂y

=∂

∂y

M

∑k=1

(λk, yk −Ak−1yk−1 + ∆thk). (A1)

Let us denote

Hk := Akyk =

ak

1yk1

ak2yk

2 + bk1yk

1...

akNyk

N + bkN−1yk

N−1

.

Then, the partial derivative (A1) can be written in the form

∂(λ, A(y, h))∂y

=∂

∂y

(M

∑k=1

(λk, yk −Hk−1)

).

By rearranging the terms and defining λM+1 = 0, we get

∂(λ, A(y, h))∂y

=∂

∂y

((λ1, H0) +

M

∑k=1

((λk, yk)− (λk+1, Hk))

).

Now,∂H0

∂y= 0 by definition of y0 and

∂Hk

∂y= Ak + Hk

1, k = 1, . . . , M− 1, where

Hk1 =

ak

11yk1 . . . ak

1Nyk1

ak21yk

2 + bk11yk+1

1 . . . ak2Nyk

2 + bk1Nyk

1...

. . ....

akN1yk

N + bkN−1,1yk

N−1 . . . akNNyk

N + bkN−1,Nyk

N−1

,

and

akij =

∂aki

∂ykj= − ∆t

∆x(g21 + g22xi)

( xj

2

)2π − ∆t

(m21 +

m22

xi+

m23

x2i

)( xj

2

)2π, (A2)

bkij =

∂bki

∂ykj=

∆t∆x

(g21 + g22xi)

( xj

2

)2π. (A3)

Thus, we can define

∂(λ, A(y, h))∂y

= (w1, w2, . . . , wM),

where

wk = λk − ((Ak)∗ + (Hk1)∗)λk+1, k = 1, . . . , M− 1

wM = λM.

Math. Comput. Appl. 2018, 23, 22 14 of 15

Then, we calculate the partial derivative of constraint (12) (implicit approximation of the stateEquation (4)) with respect to y

∂(λ, A(y, h))∂y

=∂

∂y

M

∑k=1

(λk, Bk−1yk − yk−1 + ∆thk). (A4)

Let us denote

Gk = Bk−1yk =

ak−1

1 yk1

ak−12 yk

2 − bk−11 yk

1...

ak−1N yk

N − bk−1N−1yk

N−1

.

Then, the partial derivative (A4) can be written in the form

∂(λ, A(y, h))∂y

=∂

∂y

M

∑k=1

(λk, Gk − yk−1).

Note that function Gk depends on yk−1 and yk so the partial derivative∂Gk

∂y=

∂Gk

∂yk−1 +∂Gk

∂yk .

By definition of y0, derivative∂(λ1, y0)

∂y= 0 and derivative

∂(λ1, G1)

∂y=

∂(λ1, G1)

∂y1 . By rearranging

the terms and defining λM+1 = 0, we get

∂(λ, A(y, h))∂y

=M

∑k=1

(∂(λk, Gk)

∂yk +∂(λk+1, Gk+1 − yk)

∂yk

).

Derivative∂Gk

∂yk = Bk−1 and derivative

∂Gk+1

∂yk =

ak

11yk+11 . . . ak

1Nyk+11

ak21yk+1

2 − bk11yk+1

1 . . . ak2Nyk+1

2 − bk1Nyk+1

1...

. . ....

akN1yk+1

N − bkN−1,1yk+1

N−1 . . . akNNyk+1

N − bkN−1,Nyk+1

N−1

,

where akij and bk

ij are derivatives of coefficients aki and bk

i with respect to ykj defined in

Equations (A2) and (A3), respectively. Thus, we can define

∂(λ, A(y, h))∂y

= (q1, q2, . . . , qM),

where

qk = (Bk−1)∗λk +

((∂Gk+1

∂yk

)∗− 1N

)λk+1, k = 1, . . . , M− 1,

qM = (BM−1)∗λM,

and 1N is N × N identity matrix.

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